What is the difference of a ring $R$ and the module $R_R$? It looks like they are just the same thing. Is it correct that the difference is $R$ is a ring with multiplication defined but $R_R$ is an Abelian group with addition defined only?
2026-04-05 22:17:12.1775427432
Difference of a ring and its module over itself
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Almost. It's the same thing, except that you forget the "internal" multiplication. It's more than an abelian group, because you can multiply an element of R (viewed as a module element) on the left by another element of R (viewed as a ring element), but it doesn't make sense anymore to multiply two elements of R together, viewing them both as module elements.